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Concept
Randomized rounding
Summary
Within computer science and operations research,
many combinatorial optimization problems are computationally intractable to solve exactly (to optimality).
Many such problems do admit fast (polynomial time) approximation algorithms—that is, algorithms that are guaranteed to return an approximately optimal solution given any input.
Randomized rounding
is a widely used approach for designing and analyzing such approximation algorithms.
The basic idea is to use the probabilistic method
to convert an optimal solution of a relaxation
of the problem into an approximately optimal solution to the original problem.
The basic approach has three steps:
Formulate the problem to be solved as an integer linear program (ILP).
Compute an optimal fractional solution to the linear programming relaxation (LP) of the ILP.
Round the fractional solution of the LP to an integer solution of the ILP.
(Although the approach is most commonly applied with linear programs,
other kinds of relaxations are sometimes used.
For example, see Goemans' and Williamson's semidefinite programming-based
Max-Cut approximation algorithm.)
The challenge in the first step is to choose a suitable integer linear program.
Familiarity with linear programming, in particular modelling using linear programs and integer linear programs, is required. For many problems, there is a natural integer linear program that works well,
such as in the Set Cover example below. (The integer linear program should have a small
integrality gap;
indeed randomized rounding is often used to prove bounds on integrality gaps.)
In the second step, the optimal fractional solution can typically be computed
in polynomial time
using any standard linear programming algorithm.
In the third step, the fractional solution must be converted into an integer solution
(and thus a solution to the original problem).
This is called rounding the fractional solution.
The resulting integer solution should (provably) have cost
not much larger than the cost of the fractional solution.
Official source
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